BIFURCATION THEORY AND RELATED PROBLEMS: ANTI-MAXIMUM PRINCIPLE AND RESONANCE1*

Abstract

For a smooth bounded domain Ω ⊂ IR N , we consider the b.v.p. where m ∈ L r (Ω) for some r ∈ (max {1, N/2}, + ∞], with m + ≢ 0 and g is a Carathéodory function. We deduce sufficient and sharp conditions to have subcritical (“to the left”) or supercritical (“to the right”) bifurcations (either from zero or from infinity) at an eigenvalue λ k (m) of the associated linear weighted eigenvalue problem. Furthermore, as a consequence, we also point out the bifurcation nature of some classical results like the (local) Antimaximum Principle of Clement and Peletier and the Landesman-Lazer theorem for resonant problems. In addition, we see that the bifurcation viewpoint allows to obtain also local maximum principle and more general results for some classes of strongly resonant problems. In addition, we extend the above technique to handle quasilinear b.v.p. *Supported by Acción Integrada Spain-Italy HI1997-0049, by D.G.E.S. Ministerio de Educación y Ciencia (Spain) PB98-1283 and by E.E.C. contract n. ERBCHRXCT940494. A preliminary communication of some of the results in this paper was presented at Nichtlineare Eigenwertaufgaben, held in Oberwolfach, Germany, 15–21 December 1996.